Capacitive Reactance and Impedance

1
Chapter 12

• Capacitive Reactance and Impedance

2
AC Capacitive Circuits

• In an AC capacitive circuit, the current is
  proportional to the rate of change of the
  voltage.
• The current will be greatest when the rate of
  change of the voltage is greatest, so the voltage
  and current are out of phase.
• For a capacitor, the current leads the voltage by
  90.

ICE
3
AC current and voltage in capacitors

• In an AC capacitive circuit, just like in a DC
  circuit, current is maximum when electricity is
  applied to the capacitor.
• The voltage on the capacitor when electricity is
  applied is zero.
• Since AC electricity is always changing
  direction, the voltage is always behind the
  current by 90 degrees.
• When the voltage on the capacitor is
  maximum(charged), there is no current flowing in
  the circuit however, since the voltage and
  current are always changing in an AC circuit, the
  capacitor charges and discharges.

4
Capacitors and current flow in AC circuits

• Because voltage and current are out of phase by
  90 degrees, capacitors offer an opposition to a
  change in voltage in an AC circuit.
• This opposition to a change in voltage, is based
  on the rotational speed or frequency of the AC
  sine wave applied.
• To determine speed of an object, the distance
  traveled per unit of time must be known.
• Since voltage is created by rotating a wire in a
  magnetic field, the angular speed of the rotating
  object, is based on the circumference of the
  rotation, and the number of rotations in a given
  unit of time (frequency).
• The rate of change is directly related to
  frequency of the AC sine wave. The greater the
  frequency, the higher the rate of change.

5
Angular Distance and Velocity

• Angular speed in physics is determined using the
  following formula

Where w angular speed or velocity f
frequency (cycles or rotations per
second)
6
Definition of Reactance

• Reactance is ability of a capacitor to oppose
  current flow in an AC circuit and is measured in
  ohms and the symbol is noted as XC.
• Reactance in an AC circuit, is similar to
  resistance, except that voltage and current are
  NOT in phase with each other.
• So, how is reactance calculated and what factors
  determine reactance is the next issue that needs
  to be resolved?
• Since capacitance is defined as C Q/E, we can
  mathematically derive a formula for the voltage,
  where E Q/C.
• This is an inverse relationship between the
  voltage and the capacitance of a capacitor.

7
Definition of Reactance

• Additionally, since the voltage generated is
  based on the frequency of the AC sine wave, this
  is a direct relationship and a change in the AC
  voltage is tied to the frequency of the signal.
• The math derivation for the formula can be
  simplified by combining the two relationships of
  frequency and voltage and capacitance and voltage
  with the rotational .
• Since reactance is an opposition to current flow
  in an AC circuit, Ohms Law can be modified to
  include reactance where

8
Math Derivation of Xc

• In AC since a capacitor opposes a change in
  voltage, the following equations can be derived
• DE Q/C and DE I Xc
  2p
• Substituting for DE
• I Xc 2p Q/C
• Solving for Xc
• Xc Q
• I 2p C
• Since I Q/t, substituting in the above
  equation
• Xc Q
• Q/t 2p C
• Simplifying
• Xc 1
• 1/t 2p C

9
Math Derivation of Xc

• Since 1/t f, which is frequencySimplifying
• Xc 1
• f 2p C
• Rearranging
• Xc 1
• 2p f C
• Other forms of this formulaf 1
  C 1
• 2p Xc C 2p f Xc

10
Capacitive Reactance Summary

• Capacitive reactance, XC, represents the
  opposition that capacitance presents to current
  for the AC sinusoidal case.
• XC is frequency-dependent. As the frequency
  increases, XC decreases.
• Conversely, as the frequency decreases, XC
  increases.
• A change in Xc only occurs, if the frequency of
  the AC sine wave changes.
• The effect of this relationship is that lower
  frequency AC signals tend to be blocked because
  they offer more opposition(resistance) to current
  flow.

11
Resistance

• For a resistor, the voltage and current are in
  phase.
• If the voltage has a phase angle, the current has
  the same angle.
• The impedance of a resistor is equal to R?0.

12
Capacitance

• For a capacitor, the current leads the voltage by
  90.
• If the voltage has an angle of 0, the current
  has an angle of 90.
• The impedance of a capacitor is given as XC?-90.

13
Impedance

• The resultant vector of resistance and reactance
  and the total opposition that circuit elements
  offers to current flow is the impedance, Z.
• Z Es/IT, is measured in units of ohms
• Z in polar form is Z?? where ? is the phase
  difference between the voltage and current.
• If Z has an angle of 0 degrees, there is no
  reactance in the circuit.
• IIf Z has an angle of 90 degrees, there is no
  resistance in the circuit.

14
New Terms and Quantities-1

• Admittance (symbolized as Y) the measure of how
  easily AC will flow through a circuit that
  contains a resistor and a capacitor and/or an
  inductor. It is equal to the reciprocal of the
  impedance. (1/Z)
• Conductance (symbolized as G) the measure of how
  easily AC will flow through a circuit that
  contains a resistor. It is equal to the
  reciprocal of the reistance. (1/R)
• Susceptance (symbolized as B) the measure of
  how easily AC will flow through a circuit that
  contains a capacitor or an inductor. It is equal
  to the reciprocal of the reactance. (1/X)

15
New Terms and Quantities-2

• Apparent Power (symbol Papp) the effective power
  in an AC circuit that contains resistors,
  capacitors and inductors. It is the resultant
  vector of the true power and reactive power in an
  AC circuit of the reactance. (1/X)
• Reactive Power (symbol Pvar) imaginary power or
  wattless power. Using an ideal capacitor in an
  AC circuit, no power is used. Another words all
  energy that is given by the AC source is given
  back during the discharge cycle.
• True Power (symbol Ptrue) Power used by a
  resistor. The net power is always positive in
  value and never has a negative value.

16
Power in AC capacitors circuits
In a capacitor circuit using an AC source, the
energy stored in a capacitor is given back to
the circuit during one cycle. This poweris
called reactive power, or PVAR
17
Power Formula
Parallel AC-RC Circuits
Series AC-RC Circuits
18
Solving Series RC-AC circuits

• R1 R2 C1 C2
• f 20 kHz
• Steps

19
Solving Series RC-AC circuits-2

• R1 R2 C1 C2
• f 20 kHz
• Steps

20
Vectors in Series AC-RC circuits

• Since current leads voltage by 90 in a capacitor,
  and voltage drops are present in series RC
  networks, the capacitor voltage(ECT) lags behind
  the resistor voltage(ERT) by 90 degrees.
• This means the capacitor voltage, ECT is negative
  when the ERT at 0 volts.
• Because current is constant in a series AC-RC
  circuit, reactance, resistance and impedance are
  the plotted vectors in the vector diagram
• The capacitive reactance, XCT is plotted on the
  negative (-) y axis, and the RT is plotted on the
  positive () x axis.
• The phase angle is always negative because the
  vector is in quadrant IV.

21
Solving Parallel RC-AC circuits

• R1 R2
  R3 C1 C2
• f 20 kHz
• Steps

22
Solving Parallel RC-AC circuits-2

• R1 R2
  R3 C1 C2
• f 20 kHz
• Steps

23
Vectors in Parallel AC-RC circuits

• Since current leads voltage by 90 in a capacitor,
  and branch currents are present in parallel RC
  networks, the current flowing in a capacitor(ICT)
  leads the resistor current(IRT) by 90 degrees.
• Because voltage is constant in a parallel AC-RC
  circuit, the branch currents are the plotted
  vectors in the vector diagram.
• In the vectors, the capacitor current, ICT is at
  maximum, when the resistor current, IRT is at 0
  amps.
• The capacitive current, ICT is plotted on the
  positive () y axis, and the IRT is plotted on
  the positive () x axis.
• The phase angle is always positive because the
  vector is in quadrant I.

PIC-up